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The Physics and Control
- f Balancing on a Point
The Physics and Control of Balancing on a Point Roy Featherstone - - PowerPoint PPT Presentation
The Physics and Control of Balancing on a Point Roy Featherstone - - PowerPoint PPT Presentation
The Physics and Control of Balancing on a Point Roy Featherstone 2015 Robots do not always have a polygon of support. Sometimes they have to balance actively. 2 Balancing is usually seen as a control problem, but it is also a physical
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Balancing is usually seen as a control problem, but it is also a physical process, and can be analysed as such.
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Physics of Balancing on a Point
centre
- f mass
A planar double pendulum with an actuated joint balancing on a sharp point in 2D (a knife edge in 3D). The simplest case: actuated joint passive joint
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Physics of Balancing on a Point
Maintain balance: Follow commanded motion: Objectives: 1. 2. The control problem: The controller must control 4 variables ( , , and ), but has direct control of only one variable:
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Physics of Balancing on a Point
The control solution: (in principle) If a control system succeeds in driving a variable to zero, then a side-effect is to drive , , etc. also to zero.
time
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Physics of Balancing on a Point
The control solution: (in principle) So we seek a new set of state variables to use in place of , , and with the property that controlling one has the side-effect of controlling the
- ther three.
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Physics of Balancing on a Point
Analysis: Let be the angular momentum of the robot about the support point. has the special property that is the moment of gravity about the support point.
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Physics of Balancing on a Point
Analysis: Where are elements of the joint-space inertia matrix, is the mass of the robot, and is the acceleration of gravity. Observe that and are linear functions
- f velocity.
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Physics of Balancing on a Point
Analysis: As and are linear functions of and , we can invert the equations and write where and are functions of and
- nly, and can be calculated easily via
standard dynamics algorithms.
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New Model of Balancing
The result is a new model of the balancing behaviour of the robot in which the state variables are , , and , the input is and the output is , controlling has the side-effect of maintaining the robot's balance
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New Model of Balancing
The result is a new model of the balancing behaviour of the robot in which the state variables are , , and , the input is and the output is , controlling has the side-effect of maintaining the robot's balance
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New Model of Balancing
To control the robot we map , , and to , , and , apply a simple control law to calculate , 1. 2.
- 3. convert to or as required
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Balance Controller
the gains are simple functions of , and the user's choice of poles
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Balance Controller
- ptional
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Balance Controller
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A Bit More Physics
where is the robot's natural time constant of toppling, treating it as a single rigid body is the linear velocity gain of the robot, which measures the degree to which motion of the actuated joint influences the horizontal motion of the CoM
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A Bit More Physics
A robot's velocity gain expresses the instantaneous relationship between motion
- f the actuated joint(s) and the resulting
motion of the centre of mass. where both velocity changes are caused by an impulse at joint 2. For the double pendulum,
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A Bit More Physics
disturbance response recovery disturbance response hits joint limit failure If is large then the robot is good at balancing But if is small....
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How Well Does it Work?
1 2 3 4 5 6 7 8 −1 −0.5 0.5 1 1.5 2 2.5
accurate tracking
- f linear ramp
radians seconds fast step response
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How Well Does it Work?
1 2 3 4 5 6 7 8 −1 −0.5 0.5 1 1.5 2 2.5
radians seconds non-minimum phase behaviour (but we can fix this....)
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Leaning in Anticipation
Instead of this . . . why not do this? This behaviour can be implemented by changing the command input to the controller. start responding here start leaning here
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Leaning in Anticipation
5 10 15 20 25 −1 −0.5 0.5 1 1.5 2 2.5
no anticipation anticipation
modified command
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